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Theorem ldualgrplem 29260
Description: Lemma for ldualgrp 29261. (Contributed by NM, 22-Oct-2014.)
Hypotheses
Ref Expression
ldualgrp.d  |-  D  =  (LDual `  W )
ldualgrp.w  |-  ( ph  ->  W  e.  LMod )
ldualgrp.v  |-  V  =  ( Base `  W
)
ldualgrp.p  |-  .+  =  o F ( +g  `  W
)
ldualgrp.f  |-  F  =  (LFnl `  W )
ldualgrp.r  |-  R  =  (Scalar `  W )
ldualgrp.k  |-  K  =  ( Base `  R
)
ldualgrp.t  |-  .X.  =  ( .r `  R )
ldualgrp.o  |-  O  =  (oppr
`  R )
ldualgrp.s  |-  .x.  =  ( .s `  D )
Assertion
Ref Expression
ldualgrplem  |-  ( ph  ->  D  e.  Grp )

Proof of Theorem ldualgrplem
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ldualgrp.f . . . 4  |-  F  =  (LFnl `  W )
2 ldualgrp.d . . . 4  |-  D  =  (LDual `  W )
3 eqid 2387 . . . 4  |-  ( Base `  D )  =  (
Base `  D )
4 ldualgrp.w . . . 4  |-  ( ph  ->  W  e.  LMod )
51, 2, 3, 4ldualvbase 29241 . . 3  |-  ( ph  ->  ( Base `  D
)  =  F )
65eqcomd 2392 . 2  |-  ( ph  ->  F  =  ( Base `  D ) )
7 eqidd 2388 . 2  |-  ( ph  ->  ( +g  `  D
)  =  ( +g  `  D ) )
8 eqid 2387 . . 3  |-  ( +g  `  D )  =  ( +g  `  D )
943ad2ant1 978 . . 3  |-  ( (
ph  /\  x  e.  F  /\  y  e.  F
)  ->  W  e.  LMod )
10 simp2 958 . . 3  |-  ( (
ph  /\  x  e.  F  /\  y  e.  F
)  ->  x  e.  F )
11 simp3 959 . . 3  |-  ( (
ph  /\  x  e.  F  /\  y  e.  F
)  ->  y  e.  F )
121, 2, 8, 9, 10, 11ldualvaddcl 29245 . 2  |-  ( (
ph  /\  x  e.  F  /\  y  e.  F
)  ->  ( x
( +g  `  D ) y )  e.  F
)
13 ldualgrp.r . . . . 5  |-  R  =  (Scalar `  W )
14 eqid 2387 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
154adantr 452 . . . . 5  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  ->  W  e.  LMod )
16 simpr2 964 . . . . 5  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
y  e.  F )
17 simpr3 965 . . . . 5  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
z  e.  F )
181, 13, 14, 2, 8, 15, 16, 17ldualvadd 29244 . . . 4  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( y ( +g  `  D ) z )  =  ( y  o F ( +g  `  R
) z ) )
1918oveq2d 6036 . . 3  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( x  o F ( +g  `  R
) ( y ( +g  `  D ) z ) )  =  ( x  o F ( +g  `  R
) ( y  o F ( +g  `  R
) z ) ) )
20 simpr1 963 . . . 4  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  ->  x  e.  F )
211, 2, 8, 15, 16, 17ldualvaddcl 29245 . . . 4  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( y ( +g  `  D ) z )  e.  F )
221, 13, 14, 2, 8, 15, 20, 21ldualvadd 29244 . . 3  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( x ( +g  `  D ) ( y ( +g  `  D
) z ) )  =  ( x  o F ( +g  `  R
) ( y ( +g  `  D ) z ) ) )
231, 2, 8, 15, 20, 16ldualvaddcl 29245 . . . . 5  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( x ( +g  `  D ) y )  e.  F )
241, 13, 14, 2, 8, 15, 23, 17ldualvadd 29244 . . . 4  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( ( x ( +g  `  D ) y ) ( +g  `  D ) z )  =  ( ( x ( +g  `  D
) y )  o F ( +g  `  R
) z ) )
251, 13, 14, 2, 8, 15, 20, 16ldualvadd 29244 . . . . 5  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( x ( +g  `  D ) y )  =  ( x  o F ( +g  `  R
) y ) )
2625oveq1d 6035 . . . 4  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( ( x ( +g  `  D ) y )  o F ( +g  `  R
) z )  =  ( ( x  o F ( +g  `  R
) y )  o F ( +g  `  R
) z ) )
2713, 14, 1, 15, 20, 16, 17lfladdass 29188 . . . 4  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( ( x  o F ( +g  `  R
) y )  o F ( +g  `  R
) z )  =  ( x  o F ( +g  `  R
) ( y  o F ( +g  `  R
) z ) ) )
2824, 26, 273eqtrd 2423 . . 3  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( ( x ( +g  `  D ) y ) ( +g  `  D ) z )  =  ( x  o F ( +g  `  R
) ( y  o F ( +g  `  R
) z ) ) )
2919, 22, 283eqtr4rd 2430 . 2  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( ( x ( +g  `  D ) y ) ( +g  `  D ) z )  =  ( x ( +g  `  D ) ( y ( +g  `  D ) z ) ) )
30 eqid 2387 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
31 ldualgrp.v . . . 4  |-  V  =  ( Base `  W
)
3213, 30, 31, 1lfl0f 29184 . . 3  |-  ( W  e.  LMod  ->  ( V  X.  { ( 0g
`  R ) } )  e.  F )
334, 32syl 16 . 2  |-  ( ph  ->  ( V  X.  {
( 0g `  R
) } )  e.  F )
344adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  F )  ->  W  e.  LMod )
3533adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  F )  ->  ( V  X.  { ( 0g
`  R ) } )  e.  F )
36 simpr 448 . . . 4  |-  ( (
ph  /\  x  e.  F )  ->  x  e.  F )
371, 13, 14, 2, 8, 34, 35, 36ldualvadd 29244 . . 3  |-  ( (
ph  /\  x  e.  F )  ->  (
( V  X.  {
( 0g `  R
) } ) ( +g  `  D ) x )  =  ( ( V  X.  {
( 0g `  R
) } )  o F ( +g  `  R
) x ) )
3831, 13, 14, 30, 1, 34, 36lfladd0l 29189 . . 3  |-  ( (
ph  /\  x  e.  F )  ->  (
( V  X.  {
( 0g `  R
) } )  o F ( +g  `  R
) x )  =  x )
3937, 38eqtrd 2419 . 2  |-  ( (
ph  /\  x  e.  F )  ->  (
( V  X.  {
( 0g `  R
) } ) ( +g  `  D ) x )  =  x )
40 eqid 2387 . . 3  |-  ( inv g `  R )  =  ( inv g `  R )
41 eqid 2387 . . 3  |-  ( z  e.  V  |->  ( ( inv g `  R
) `  ( x `  z ) ) )  =  ( z  e.  V  |->  ( ( inv g `  R ) `
 ( x `  z ) ) )
4231, 13, 40, 41, 1, 34, 36lflnegcl 29190 . 2  |-  ( (
ph  /\  x  e.  F )  ->  (
z  e.  V  |->  ( ( inv g `  R ) `  (
x `  z )
) )  e.  F
)
431, 13, 14, 2, 8, 34, 42, 36ldualvadd 29244 . . 3  |-  ( (
ph  /\  x  e.  F )  ->  (
( z  e.  V  |->  ( ( inv g `  R ) `  (
x `  z )
) ) ( +g  `  D ) x )  =  ( ( z  e.  V  |->  ( ( inv g `  R
) `  ( x `  z ) ) )  o F ( +g  `  R ) x ) )
4431, 13, 40, 41, 1, 34, 36, 14, 30lflnegl 29191 . . 3  |-  ( (
ph  /\  x  e.  F )  ->  (
( z  e.  V  |->  ( ( inv g `  R ) `  (
x `  z )
) )  o F ( +g  `  R
) x )  =  ( V  X.  {
( 0g `  R
) } ) )
4543, 44eqtrd 2419 . 2  |-  ( (
ph  /\  x  e.  F )  ->  (
( z  e.  V  |->  ( ( inv g `  R ) `  (
x `  z )
) ) ( +g  `  D ) x )  =  ( V  X.  { ( 0g `  R ) } ) )
466, 7, 12, 29, 33, 39, 42, 45isgrpd 14757 1  |-  ( ph  ->  D  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {csn 3757    e. cmpt 4207    X. cxp 4816   ` cfv 5394  (class class class)co 6020    o Fcof 6242   Basecbs 13396   +g cplusg 13456   .rcmulr 13457  Scalarcsca 13459   .scvsca 13460   0gc0g 13650   Grpcgrp 14612   inv gcminusg 14613  opprcoppr 15654   LModclmod 15877  LFnlclfn 29172  LDualcld 29238
This theorem is referenced by:  ldualgrp  29261
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-n0 10154  df-z 10215  df-uz 10421  df-fz 10976  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-plusg 13469  df-sca 13472  df-vsca 13473  df-0g 13654  df-mnd 14617  df-grp 14739  df-minusg 14740  df-sbg 14741  df-cmn 15341  df-abl 15342  df-mgp 15576  df-rng 15590  df-ur 15592  df-lmod 15879  df-lfl 29173  df-ldual 29239
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